Mixed Radial Volume Comparison under Spectral Ricci Bounds

Wed, 02 Sep 2026 00:00:00 +0000

Main results

This paper develops radial volume comparison under a spectral lower bound for Ricci curvature. The mixed radial construction separates the conformal geometry used to choose rays from the weights used to truncate and measure them.

Sharp comparison

A space-form model for mixed radial balls

Under the spectral inequality $\operatorname{Ric}_g-\alpha(\Delta_g u/u)g\geq(n-1)\kappa g$, the weighted volume of the $g$-arclength mixed radial domain satisfies a scale-invariant comparison with the corresponding space-form volume.

Polynomial growth

No pointwise bounds on the spectral weight

For an explicit range of radial and measure exponents, mixed radial balls have polynomial weighted-volume growth. The optimal exponent can attain the Euclidean value $n$.

Applications

Diameter, sharp volume, and stable Bernstein

The comparison gives radial proofs of the spectral Bonnet--Myers and sharp volume theorem of Antonelli--Xu, together with a short volume-growth proof that every complete two-sided stable minimal immersion $M^3\to\mathbb R^4$ is flat.

Proof architecture

The argument begins with an exact Riccati identity for the weighted radial Jacobian along geodesics of the conformal metric $h_\alpha=u^{2\alpha}g$. Taking the $(n-1)$-st root of this Jacobian converts the identity into a one-dimensional Sturm comparison and yields the space-form estimate. For general mixed radial balls, the logarithmic derivatives of the Jacobian and the spectral weight are instead treated as coupled variables; a quadratic estimate and a Bellman-type correction produce the polynomial-growth monotonicity formula. The positive-curvature case also gives an optimal diameter bound among smooth closed examples.

Volume Comparison | Gaoming Wang

Mixed Radial Volume Comparison under Spectral Ricci Bounds

Main results

A space-form model for mixed radial balls

No pointwise bounds on the spectral weight

Diameter, sharp volume, and stable Bernstein

Proof architecture